Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
749 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2862 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15253 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 38. C2×#97 | Q8 | semidirect |
| 39. C2×C4×C8 | Q8 | semidirect |
| 45. #376 | Q8 | semidirect |
| 47. C43 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21330 | 2 |
| #11389 | 2 |
| #11349 | 2 |
| #10816 | 2 |
| #10805 | 2 |
| C4×H(3,ℤ/4) | 2 |